Yes, traditionally St Nicholas puts sweets in children's shoes on 6th December. However, this has never really been a British tradition.
We have Father Christmas, who was more about getting pissed and living it large. The puritans didn't really approve of him, and he has come in and out of favour. The Scottish presbytarians really weren't that keen on him - and the big midwinter celebration in Scotland has been Hogmany. Yes he is probably based on the idea of the 'Green Man', who may have been based on earlier Gods, but I think that is getting quite far away from Santa and Rudolph.
Father Christmas and Santa were merged by the Victorians and the current concept of Christmas evolved in the second half of the twentieth century.
I really, really don't think that children of my grandparents generation were that bothered either way about Santa Claus, and children of my great grandparents generation would probably be a bit confused. (Particularly as some of my great grandparents were quite hard line Scottish presbytarians, and others would have probably been more interested in getting something to eat and a warm bed on Christmas Day).
I am now proud to live in a society where I have no idea what the personal beliefs or traditions of any person might be. I have no wish for my children to tell anybody that their beliefs are a load of rubbish as that is just rude. However the idea that there is some truth/secret about Santa Claus that we all have decide to tell/not tell to our children and then swear them to secrecy and that this is a foundational part of British society is really a bit far fetched.
My 4 year old might tell your 4 year old that there is no Santa, and he might also say that his dad is an astronaut. In both situations you can say "really?" and get on with your day.
Getting cross with me for ruining your child's Christmas would be just plain silly.